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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 139583, 24 pagesdoi:10.1155/2012/139583

Research ArticleA Stabilized Incompressible SPH Method byRelaxing the Density Invariance Condition

Mitsuteru Asai,1 Abdelraheem M. Aly,1 Yoshimi Sonoda,1and Yuzuru Sakai2

1 Department of Civil Engineering, Kyushu University, 744 Motooka, Nishi–ku, Fukuoka 819-0395, Japan2 Faculty of Education and Human Science, Yokohama National University, 79-1 Tokiwadai, Hodogaya-ku,Yokohama 240-8501, Japan

Correspondence should be addressed to Mitsuteru Asai, [email protected]

Received 5 January 2012; Accepted 16 March 2012

Academic Editor: Hiroshi Kanayama

Copyright q 2012 Mitsuteru Asai et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A stabilized Incompressible Smoothed Particle Hydrodynamics (ISPH) is proposed to simulatefree surface flow problems. In the ISPH, pressure is evaluated by solving pressure Poisson equationusing a semi-implicit algorithm based on the projection method. Even if the pressure is evaluatedimplicitly, the unrealistic pressure fluctuations cannot be eliminated. In order to overcome thisproblem, there are several improvements. One is small compressibility approach, and the other isintroduction of two kinds of pressure Poisson equation related to velocity divergence-free anddensity invariance conditions, respectively. In this paper, a stabilized formulation, which wasoriginally proposed in the framework of Moving Particle Semi-implicit (MPS) method, is appliedto ISPH in order to relax the density invariance condition. This formulation leads to a new pressurePoisson equation with a relaxation coefficient, which can be estimated by a preanalysis calculation.The efficiency of the proposed formulation is tested by a couple of numerical examples of dam-breaking problem, and its effects are discussed by using several resolution models with differentparticle initial distances. Also, the effect of eddy viscosity is briefly discussed in this paper.

1. Introduction

The meshless particle methods have been applied in many engineering applications includ-ing the free-surface fluid flows. In the particle methods, the state of a system is representedby a set of discrete particles, without a fixed connectivity; hence, such methods are inherentlywell suited for the analysis of moving discontinuities and large deformations such as thefree-surface fluid flows with breaking and fragmentation.

The SPH technique was originally proposed by Lucy [1] and further developedby Gingold and Monaghan [2] for treating astrophysical problems. Its main advantage is

2 Journal of Applied Mathematics

the absence of a computational grid or mesh since it is spatially discretized into Lagrangianmoving particles. This allows the possibility of easily modeling flows with a complexgeometry or flows where large deformations or the appearance of a free surface occurs. At thepresent time, it is being exploited for the solution of problems appearing in different physicalprocesses. Monaghan [3] has provided a fairly extensive review of SPH methods. The SPHmethod had been applied into compressible and incompressible viscous flow problems [4–7].The SPH is originally developed in compressible flow, and then some special treatment isrequired to satisfy the incompressible condition. One approach is to run the simulations in thequasi-incompressible limit, that is, by selecting the smallest possible speed of sound whichstill gives a very low Mach number ensuring density fluctuations within 1% [4, 5]. Thismethod is known as the Weakly Compressible Smooth Particle Hydrodynamics (WCSPH).In the WCSPH, the artificial viscosity, which is originally developed by Monaghan, has beenwidely used not only for the energy dissipation but also for preventing unphysical penetra-tion of particles. Recently, a proposal for constructing an incompressible SPHmodel has beenintroduced, whose pressure is implicitly calculated by solving a discretized pressure Poissonequation at every time step [8–17].

Lee et al. [13] presented comparisons of a semi-implicit and truly incompressible SPH(ISPH) algorithm with the classical WCSPH method, showing how some of the problemsencountered in WCSPH have been resolved by using ISPH to simulate incompressible flows.They used the function of temporal velocity divergence for discretized source terms of Pois-son equation of pressure to ensure truly incompressible flow. Khayyer et al. [14, 15] proposeda corrected incompressible SPH method (CISPH) derived based on a variational approach toensure the angular momentum conservation of ISPH formulations to improve the pressuredistribution by improvement of momentum conservation and the second improvement isachieved by deriving and employing a higher-order source term based on a more accuratedifferentiation.

The source term in pressure Poisson equation (PPE) for ISPH is not unique; it has sev-eral formulations in the literature; one of them as a function of density variation and the otherutilizes velocity divergence condition to formulate the source term. The former formulationwith the density variation can keep a uniform particle distribution, although evaluatedpressure include high unrealistic fluctuation. On the other hand, the formulation of thedivergence-free condition evaluates much smoother pressure distribution, but density errorsmay occur due to particle clustering. Then, modified schemes have been proposed to satisfythe above two conditions: density invariant and divergence-free condition. Pozorski andWawrenczuk [9] proposed a modified scheme, in which both the PPEs are solved separatelyat two intermediate states in each time step. Hu and Adams [16, 17] introduced internal itera-tions to satisfy both conditions accurately at the samemoment. These modified schemes needto solve multiple PPEs in each time step, and these computational costs become expensecompared to the conventional ISPH.

Recently, in the framework of MPS, there is a trend to introduce a higher-order sourceterm in the PPE. Kondo and Koshizuka [18] proposed a new formulation with a source termcomposed by three parts; one is main part and another two terms related to error-compensating parts. Tanaka and Masunaga [19] introduced a similar high-order source termwith two components incorporated with quasi-compressibility. Note that the number of PPEsper time step in their higher-order source term formulations is just one and its numerical costis almost same as the original scheme. In this study, we reformulate a source term of the PPEwhich contains both contributions from velocity-divergence-free and density invarianceconditions. Only one PPE per time step should be solved as the recent development in

Journal of Applied Mathematics 3

the MPS, but our formulation with a relaxation coefficient is unique. Note that, the relaxationcoefficient depends on the initial particle distance, and a suitable relaxation coefficient canbe obtained from the hydrostatic pressure calculations as a preanalysis. The accuracy andthe efficiency of the proposed model are investigated in a couple of examples which arepreviously selected in published papers.

The turbulence models in the SPH are also important issue and the effects in theWCSPH have been nicely investigated by Violeau and Issa [20]. Lee et al. have introduced thesame turbulence model such as k-ε model into ISPH. Gotoh et al. [21] and Shao and Gotoh[22] introduced the static Smagorinsky model into the ISPH, and he discussed the effect ofadditional eddy viscosity shortly. In this paper, we also discuss the effect of eddy viscosityfrom our simulation results.

2. Typical Incompressible Smoothed ParticleHydrodynamics (ISPHs) Formulation

In this section, typical ISPH formulation, which is similar procedure in moving particle semi-implicit method (MPS) proposed by Koshizuka and Oka [24], is summarized. The mainfeature is that semi-implicit integration scheme is applied into particle discretized equationsfor the incompressible flow problem. The original idea of the semi-implicit scheme is calledby projection method, which has been widely used in the finite difference method and in thefinite element method. After the basic application of projection method into SPH is describedhere, several similar schemes will be categorized by the difference of treatment of PPE in thenext section.

2.1. The Governing Equations for Incompressible Flow

In the Lagrange description, the continuity equation and the Navier-Stokes equations can bewritten as

Dρ

Dt+ ρ∇ · u = 0, (2.1)

DuDt

= −1ρ∇P + υ∇2u + g +

1ρ∇ · τ , (2.2)

where ρ and υ are density and kinematic viscosity of fluid, u and P are a velocity vector andpressure of fluid, respectively, g is gravity acceleration, and t indicates time. The turbulencestress τ is necessary to represent the effects of turbulence with coarse spatial grids, and itsapplication into the particle simulation has been initially developed by Gotoh et al. [21]. Inthe most general incompressible flow approach, the density is assumed by a constant valuewith its initial value ρ0. Then, the aforementioned governing equations lead to

∇ · u = 0, (2.3)

DuDt

= − 1ρ0

∇P + υ∇2u + g +1ρ0

∇ · τ . (2.4)


2.2. Projection Method

In the projection method [25], the velocity-pressure-coupled problem has been solved sepa-rately for velocity and pressure. Here, all the state variables may update from a previous timestep to current time step. Below, superscripts (n) and (n + 1) indicate previous and currenttime step, respectively. In the first predictor step, intermediate state without pressure gradientis assumed, and its velocity field is indicated by u∗. The intermediate velocity field can beevaluated by solving the following equation:

u∗ − un

Δt= υ∇2un + g +

1ρ0

∇ · τ , (2.5)

(Predictor) : u∗ = un + Δt

(υ∇2un + g +

1ρ0

∇ · τ), (2.6)

Then, the following corrector step introduces an effect of remaining “current” pressuregradient term as follows:

un+1 − u∗

Δt= − 1

ρ0∇Pn+1, (2.7)

(Corrector) : un+1 = u∗ + Δu∗ = u∗ −Δt

(1ρ0

∇Pn+1), (2.8)

where Δu∗ indicates the incremental velocity from the predicted velocity u∗.The key point here is the evaluation of “current” pressure value. By taking the

divergence of correction step (2.7) as

∇ ·(

un+1 − u∗

Δt

)= −∇ ·

(1ρ0

∇Pn+1). (2.9)

Then, the incompressible condition (2.3) leads to

∇ · un+1

Δt= 0. (2.10)

By substituting (2.10) into (2.9), this leads to the following pressure Poisson equation (PPE):

∇2Pn+1 = ρ0∇ · u∗

Δt. (2.11)

The above corrector step can be implemented by substituting the pressure gradient with thesolution of PPE.


2.3. The SPH Methodology

A spatial discretization using scattered particles, which is based on the SPH, is summarized.First, a physical scalar function φ(xi, t) at a sampling point xi can be represented by thefollowing integral form:

φ(xi, t) =∫W(xi − xj , h

)φ(xj , t)dv =

∫W(rij , h

)φ(xj , t)dv, (2.12)

where W is a weight function called by smoothing kernel function in the SPH literature. Inthe smoothing kernel function, rij = |xi−xj | and h are the distance between neighbor particlesand smoothing length, respectively. For SPH numerical analysis, the integral equation (2.12)is approximated by a summation of contributions from neighbor particles in the supportdomain.

φ(xi, t) ≈⟨φi

⟩=∑j

mj

ρjW(rij , h

)φ(xj , t), (2.13)

where the subscripts i and j indicate positions of labeled particle, and ρj andmj mean densityand representative mass related to particle j, respectively. Note that the triangle bracket 〈φi〉means SPH approximation of a function φ. The gradient of the scalar function can be assumedby using the above defined SPH approximation as follows:

∇φ(xi) ≈⟨∇φi

⟩=

1ρi

∑j

mj

(φj − φi

)∇W(rij , h

). (2.14)

Also, the other expression for the gradient can be represented by

∇φ(xi) ≈ 〈∇φi〉 = ρi∑j

mj

(φj

ρ2j+φi

ρ2i

)∇W

(rij , h

). (2.15)

In this paper, quintic spline function [26] is utilized as a kernel function.

W(rij , h

)= βd

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3 − rij

h

)5

− 6(2 − rij

h

)5

+ 15(1 − rij

h

)5

, 0 ≤ rij < h,

(3 − rij

h

)5

− 6(2 − rij

h

)5

, h ≤ rij < 2h,(3 − rij

h

)5

, 2h ≤ rij < 3h,

0, rij ≥ 3h,

(2.16)

where βd is 7/478πh2 and 3/358πh3, respectively, in two- and three-dimensional space. Ithas been observed that a cubic spline produces fluctuations in the pressure and velocityfields for fluid dynamics simulation, and the quintic spline shown in (2.16) gives more stablesolutions.


2.4. Projection-Based ISPH Formulations

Here, the projection method for incompressible fluid problem, which is summarized inSection 2.2, is descretized into particle quantities based on the SPH methodology. For thispurpose, the gradient of pressure and the divergence of velocity are approximated as follows:

∇p(xi) ≈⟨∇pi

⟩= ρi∑j

mj

(pj

ρ2j+

pi

ρ2i

)∇W

(rij , h

), (2.17)

∇ · u(xi) ≈ 〈∇ · ui〉 =1ρi

∑j

mj

(uj − ui

) · ∇W(rij , h

). (2.18)

Although the Laplacian could be derived directly from the original SPH approximation of afunction in (2.17), this approach may lead to a loss of resolution. Then, the second derivativeof velocity for the viscous force and the Laplacian of pressure have been proposed by Morriset al. [5] by an approximation expression as follows:

∇ · (υ∇u)(xi) ≈ 〈∇ · (υ∇ · ui)〉 =∑j

mj

(ρiυi + ρjυj

ρiρj

rij∇W(∣∣ri − rj

∣∣, h)r2ij + η2

)uij , (2.19)

where η is a parameter to avoid a zero dominator, and its value is usually given by η2 =0.0001h2. For the case of υi = υj and ρi = ρj , the Laplacian term is simplified as

〈∇ · (υ∇ · ui)〉 =2υi

ρi

∑j

mj

(rij∇W

(∣∣ri − rj∣∣, h)

r2ij + η2

)uij . (2.20)

Similarly, the Laplacian of pressure in pressure Poisson equation (PPE) is given by

∇2p(xi) ≈⟨∇2pi

⟩=

2ρi

∑j

mj

(pijrij∇W

(∣∣ri − rj∣∣, h)

r2ij + η2

). (2.21)

The PPE after SPH interpolation is solved by a preconditioned (diagonal scaling) ConjugateGradient (PCG)method [27] with a convergence tolerance (= 1.0 × 10−9).

2.5. Modeling of the Turbulence Stress

When dealing with the turbulent flows, the turbulent stress in (2.2), which are called by sub-particle scale stress in the particle simulations, needs to be modeled. In this paper, a largeeddy simulation approach [21, 22] is used for modeling the turbulent stress as

τIJρ0

= 2υTSIJ − 23kδIJ , (2.22)

where υT and k are the turbulence eddy viscosity and the turbulence kinetic energy, res-pectively. SIJ indicates the strain rate of the mean flow, and δIJ is the Kronecker delta. It is


assumed in this paper that the eddy viscosity is modeled by the static Smagorinsky modelas υT = (CsΔ)2|S|, in which Cs = 0.2 is the Smagorinsky constant (taken as the analyticalvalue in this paper), the constantΔ is taken as 2h, in which h is the smoothing length definedin (2.12). The local strain rate |S| = (2SIJSIJ)

1/2 can be calculated in the SPH formulation asVioleau and Issa [20].

2.6. Treatment of No-Slip Boundary Condition

The boundary condition on the rigid bodies has an important role to prevent penetration andto reduce error related to truncation of the kernel function. Takeda et al. [28] andMorris et al.[5] have introduced a special wall particle which can satisfy imposed boundary conditions.Recently, Bierbrauer et al. [29] described a consistent treatment of boundary conditions,utilizing the momentum equation to obtain approximations to velocity of image particles.

In our research, dummy particles technique, in which dummy particles are regularlydistributed at the initial state and have zero velocity through the whole simulation process,is utilized just for simplicity in the implementation. In the following simulation, the pressurePoisson equation is solved for all particles including these dummy particles to get an enoughrepulsive force preventing penetration.

3. Stabilizations of Pressure Evaluation in Pressure Poisson Equation

Here, the pressure Poisson equation is reconsidered to overcome the error of artificialpressure fluctuation in the ISPH. The key points are related to the accuracy of densityrepresentation in SPH formulation and the treatment of pressure Poisson equation.

3.1. Keeping Divergence Free Scheme

Divergence free condition in the projection-based ISPH has been initially proposed by Cum-mins and Rudman [8]. Lee et al. [13] has applied it into the Reynolds turbulencemodel whichuses an averaging in time. They called by “truly” incompressible SPH since the initial densityρ0 is assumed constant for each particle. Then the divergence of the intermediate velocity hasbeen used to calculate the PPE as mentioned above in (2.11). The PPE can be written in SPHapproximation by substituting (2.18) and (2.21) as follows:

⟨∇2pn+1i

⟩=

ρ0

Δt

⟨∇ · u∗i

⟩. (3.1)

3.2. Keeping Density Invariance Scheme

The alternative scheme can be derived by using a density invariance condition [14, 15]. Here,the “particle” density in the SPH is evaluated by

ρ(xi) ≈⟨ρi⟩=∑j

mjW(rij , h

)=∑j

mjWij . (3.2)

The particle position updates after each predictor step in the density invariance schemeand the particle density is updated on the intermediate particle positions. The intermediate


particle density is indicated by 〈ρ∗i 〉. By assuming incompressibility condition with 〈ρn+1i 〉 =ρ0, the mass conservation law (2.1) can be rewritten for each particle as follows:

⟨ρ∗i⟩ − ρ0

Δt+ ρ0⟨∇ · u∗

i

⟩= 0. (3.3)

By substituting (3.3) into (2.11) and using the SPH form, the PPE for ISPH can beapproximately redefined by

〈∇2pn+1i 〉 =ρ0 − 〈ρ∗i 〉

Δt2. (3.4)

The main difference between the keeping divergence-free and keeping density-invariancescheme appeared in the source term of the PPE. Note that this keeping density-invariancescheme is analogous to the formulation in the MPS, although the MPS utilizes a “particlenumber” density instead of the particle density. The above two schemes have a relationship.Ataie-Ashtiani and Shobeyri [30] has converted from a PPE in the keeping density invariancescheme to a PPE in the keeping divergence-free scheme.

3.3. Combination Scheme of Both Divergence-Free andDensity-Invariance Condition

A notable scheme was proposed by Hu and Adams [16]. The divergence-free and density-invariance conditions are sufficiently satisfied in their scheme. As they discussed, thedivergence-free scheme calculates a smoothed pressure field but a large density variation willappear. Hu and Adams’s scheme includes an internal iteration at each step, and two kinds ofPPEs should be solved until both the divergence-free and density-invariance conditions areapproximately satisfied. According to Xu et al. [31], this scheme shows accurate and robustsolutions, but total calculation time shows 4-5 times higher than that of the above two scheme.

3.4. Relaxed Density Invariance Scheme Incorporated withDivergence-Free Condition

Here, we proposed an efficient and robust ISPH scheme using both conditions withoutinternal iterations. In the sense of physical observation, physical density should keep its initialvalue for incompressible flow. However, during numerical simulation, the “particle” densitymay change slightly from the initial value because the particle density is strongly dependenton particle locations in the SPH method. If the particle distribution can keep almost uni-formity, the difference between “physical” and “particle” density may be vanishingly small.In other words, accurate SPH results in incompressible flow need to keep the uniform particledistribution. For this purpose, the different source term in pressure Poisson equation can bederived using the “particle” density. The SPH interpolations are introduced into the originalmass conservation law before the perfect compressibility condition is applied

⟨∇ · un+1

i

⟩= − 1

ρ0

⟨ρn+1i

⟩ − ⟨ρni ⟩Δt

. (3.5)


By substituting (3.5) into (2.9) and using SPH form, the PPE can be represented by

⟨∇2pn+1i

⟩=

ρ0

Δt

⟨∇ · u∗i

⟩+

⟨ρn+1i

⟩ − ⟨ρni ⟩Δt2

. (3.6)

Here, it is assumed that the current particle density is “hopefully” closed to initialdensity and the incremental particle density 〈Δρni 〉 are defined by

⟨ρn+1i

⟩=⟨ρni⟩+⟨Δρni⟩ ≈ ρ0, (3.7)

⟨Δρni⟩:= α(ρ0 − ⟨ρni ⟩

), (3.8)

where the above integration scheme is called by the method of coordinate descent witha relaxation coefficient α (0 ≤ α ≤ 1), and the PPE is modified as follows:

⟨∇2pn+1i

⟩=

ρ0

Δt

⟨∇ · u∗i

⟩+ α

ρ0 − ⟨ρni ⟩Δt2

. (3.9)

The similar equation, in which the density is replaced by a particle number density,was proposed by Losasso et al. [32], but they did not introduce the relaxation coefficient.Note that our proposed scheme couples the divergence-free and a relaxed density-invariancecondition, and a special case using α = 0 leads to the original divergence-free scheme. Theeffect of the relaxation coefficient will be tested in the later examples. Figure 1 shows the flowcharts of these schemes to show the difference between existing and our proposed scheme.Similar modifications in the source term of PPE have been proposed in the MPS by Tanakaand Masunaga [19]. Recently, Khayyer and Gotoh [33] proposed a different higher-ordersource term without the unknown coefficient like the relaxation coefficient in this paper. It isimportant that the relaxation coefficient is strongly dependent on the initial particle distance,and the optimum value can be calibrated by a simple hydrostatic pressure test with the sameinitial particle distance as the final simulation model. The hydrostatic pressure test is calledby preanalysis in this paper.

3.5. Tracking the Free Surface Boundary

Detection of free surface has an important role in the ISPH for free surface flow, becausethe pressure values on free surface particles should be equal to zero as Dirichlet boundaryconditions of PPE. The method how to track the free surface may differ in each ISPH scheme.

Usually in the keeping density-invariance scheme, surface particles have beendetected by referring the current particle density 〈ρi〉. The details have been discussed byGotoh et al. [21], Shao and Gotoh [22], and Khayyer et al. [14, 15]. On the other hand, in thekeeping divergence-free scheme, Lee et al. [13] proposed a new treatment with the divergenceof a particle position vector. If the particle density keeps around its initial value, the formerfree surface detection method can be utilized. In our simulation, surface particles are simplyjudged by the total number of neighboring particles.

G. R. Liu and M. B. Liu [34] have investigated the number of neighboring particlesto estimate an efficient variable smoothing length for the adaptive analysis. In the case of


Corrector

• Update velocity and position

• Solve a pressure Poisson eqn.

Start

Input initial coordinateand boundary condition

PredictorEvaluate intermediate velocity

Check END

End

Corrector

• Update velocity, position and density

• Solve a pressure Poisson eqn.

Start

Input initial coordinateand boundary condition

Predictor• Evaluate intermediate velocity

• Update position and density

Check END

End

Corrector

• Update velocity, position and density

• Solve a pressure poisson eqn.

Start

Input initial coordinate and boundary condition

Predictor• Evaluate intermediate velocity

Check END

End

In th

e ne

xt s

tep

In th

e cu

rren

t ste

p

= =∇ · u∗i

ρ0

∆tρ0 − ρ∗i

∆t2

n = n + 1 n = n + 1 n = n + 1

∇2Pn+ +1i

=ρ0

∆t∇ · u∗

iαρ0 − ρni

∆t2

•

∇2Pn+1i ∇2Pn+1

i

(a) keeping divergence-free scheme (b) keeping density-inveriance scheme (c) relaxed density-inveriance scheme(= proposed scheme)

Figure 1: Flow charts in the ISPH schemes.

16

12

8

4

−1

−5−5 1 7 13 19 25

Point A

d0 = 0.01 m

16

12

8

5

1

−3−3 3 8 13 18 23

Point A

d0 = 0.005 m

15

12

9

5

2

−1−1 3 8 12 17 21

Point A

d0 = 0.0025 m

Figure 2: Schematic diagram of hydrostatic pressure at point A for three values of particle sizes.

a simply cubic patterned lattice, h is usually chosen as larger than 1.2 times of the initialparticle distance d0. They showed that the number of neighboring particle within the supportdomain khwith k = 2 for cubic spline kernel function should be about 21 in two-dimensionalsimulations.We checked a threshold for judging free surface particles for quintic spline kernelfunction with k = 3, and the threshold should be about 28 and 190 in 2D and 3D, respectively.

4. Preanalysis to Determine an Efficient Relaxation Coefficient

In this section, hydrostatic pressure evaluations are performed to investigate the effects ofrelaxation coefficient and to determine a suitable range of its value with reference to an initialparticle distance.

The three particle models have been generated with different initial particles distancesd0 = 0.01, 0.005, and 0.0025m as shown in Figure 2. The theoretical hydrostatic pressure isgiven by a law: p = ρgh (= 980N/m2) with water density ρ = 1000 kg/m3 and a waterheight h = 0.1m. Figure 3 shows pressure histories with different relaxation coefficients for


2 4 6 8200

300

400

500

600

700

800

900

1000

1100

1200

Hydrostatic pressure

Pres

sure

(Pa)

Time (s)

α = 0.001α = 0.01

α = 0.1α = 0.2

(a) At particle size d0 = 0.01m

2 4 6 8200

300

400

500

600

700

800

900

1000

1100

1200


Pres

sure

(Pa)

Time (s)

α = 0.00025α = 0.0005

α = 0.001α = 0.002

(b) At particle size d0 = 0.005m

2 4 6 8200300400500600700800900

100011001200130014001500


Pres

sure

(Pa)

Time (s)

α = 0.000025α = 0.00005

α = 0.0001α = 0.00025

(c) At particle size d0 = 0.0025m

Figure 3: Time history for pressure distributions under the effect of relaxation coefficient at differentparticle size models d0 = 0.01, 0.005, and 0.0025m, respectively.

each model. From this figure, the proper ranges of relaxation coefficient αwith initial particledistances d0 = 0.01, 0.005, and 0.0025m are approximately about (0.1 : 0.2), (0.0005 : 0.002)and (0.00005 : 0.0002), respectively. In this paper, a constant time is chosen by Δt = 0.1d0

corresponding to Khayyer et al. [15]. Here note that the optimum parameter calibrated fromthis preanalysis can use the same value in the later examples if themodel has the same particleresolution.

5. Numerical ExamplesHere, several numerical examples are solved by the current scheme with an efficient relax-ation coefficient, which are calibrated by the hydrostatic pressure evaluation in the previoussection.


5.1. The Effect of Relaxation Coefficient during Dam Break Simulation

A two-dimensional dam break analysis is performed to compare the proper relaxationcoefficient between hydrostatic pressure and dam break simulation with the same particledistance. The geometry of a 2D dam break is shown in Figure 4, where the particle distanced0 = 0.01m, the water width L = 0.20m, water height Hw = 2.5L, and the wall widthWl = 5 L.

At first, Figure 5 shows the results of free surface detection by using the number ofneighbors. It is seen that this simple free surface detection scheme is sufficiently accurate todetermine the Dirichlet boundary conditions of the pressure Poisson equation and it is alsosuitable for any formulation of the pressure Poisson equation. The effects of relaxation coeffi-cient are investigated by the density errors. Two boundary particles A and B, which positionis marked in Figure 4, are selected to output an evaluated numerical density. Figure 6 showsthe time histories of the density at particles A and B. From this observation with a particledistance d0 = 0.01m, it seems that, too low relaxation coefficient (below 0.01) the densityerrors are high. A proper range of relaxation coefficient α = 0.1∼0.25 leads to a stable solution.In addition, the density error fluctuations become serious when the relaxation coefficient islarger than this proper range. In the same way, the suitable ranges of relaxation coefficientfor different particles distances d0 = 0.005, 0.0025m are evaluated by (0.0005 : 0.0025) and(0.00005 : 0.0001), respectively. Note that these proper ranges for each initial particle distanceare close to the preevaluated proper ranges calibrated with the hydrostatic pressure test.Finally, the optimum values of relaxation coefficient in 2D dam break analysis for particlesizes d0 = 0.01, 0.005, and 0.0025m are determined by 0.15, 0.001, and 0.00006, respectively.

Figure 7 shows the pressure distributions for three models with different initialparticle distances d0 = 0.01, 0.005, and 0.0025m. A suitable relaxation coefficient is utilizedfor each model. The first water impact at the right wall generates highest pressure, and itreturns in the form of a jet. Then, it becomes a stable state after two more water impacts acton both side walls. The snapshots for water impact, after the first water impact, reversing jetand water stable state are captured from each model with different particle distance. Thesesnapshots at the same time show similar water shape. The pressure histories at the rightcorner are plotted in Figure 8. Although unrealistic pressure fluctuation appears in the caseof lowest resolution model with d0 = 0.01m, a similar tendency of pressure history can getfrom different resolution models. Adjusting suitable relaxation coefficient can increase thepressure smoothness. The hydrostatic pressure after this dam break analysis is analyticallyevaluated by 1000N/m2, and our evaluated pressure after 4 seconds looks to converge intothe analytical hydrostatic value. In this example, water front speed is plotted in Figure 9. Ourresults shows a good agreement with the experimental results obtained by Koshizuka andOka [24] and Martin and Moyce [35], moreover the numerical results obtained by Lee et al.[23].

5.2. Comparison of Configurations and Pressure duringDam Break Simulation

Next, water configurations and pressure distribution are comparedwith an experimental databy Zhou et al. [36] and with a result by original incompressible SPH, which is the same asa special case of our proposed scheme with α = 0. The schematic diagram is the same asZhou’s experiment is shown in Figure 10, and the pressure measuring point is located at


L = 20 cm

Hw = 2.5L

5LL = 20 cm

HwHH = 2.5L

5L57

44

32

19

7

−6−6 16 38 61 83 105

Figure 4: Dam break analysis (unit: cm).

Free surface particlesInner particles

t = 0.25 s

t = 0.95 s

t = 1 s

Figure 5: Detect free surface numerically using the number of neighbouring particles for using k = 3 andh = 1.2d0 for quintic spline kernel function at initial particle size d0 = 0.005m.


0 2 4 6 8−5

0

5

10

15

20

25

30

35

40

Time (s)

Den

sity

err

or (%

)

(a) At particle A

0 2 4 6 8

−10

0

10

20

30

40

Den

sity

err

or (%

)

Time (s)

α = 0.01α = 0.15α = 0.25

(b) At particle B

Figure 6: Effect of relaxation coefficient on the density evaluation error at (a) particles A and (b) ParticleB, respectively.

a point on the right wall (0.16m). The particle initial distance is selected as d0 = 0.005m.A proper relaxation coefficient for this resolution is selected by α = 0.001 which is the sameoptimum value in the hydrostatic pressure test, and then the numerical solution is comparedto the truly incompressible scheme with α = 0. Figure 11 shows the comparison results ofthe snapshots with pressure distribution from the initial state to the final stable state. Thesnapshots from each scheme are captured at the first water impact, running up along the rightwall reversing development of splash-up and the stable state. Although the waveconfigurations show similarities, the pressure value from the truly incompressible schemeis less than that from our proposed scheme. In addition, the total volume of the water at


t = 0.4 st = 0.4 s t = 0.4 s

t = 0.5 st = 0.5 st = 0.5 s

t = 0.9 st = 0.9 st = 0.9 s

t = 1 st = 1 st = 1 s

t = 2.5 st = 2.5 st = 2.5 s

t = 4 s t = 4 st = 4 s

0 500 1000 1500 2000 2500(Pa)

0 500 1000 1500 2000 2500(Pa)

0 500 1000 1500 2000 2500(Pa)

(a) Particle size d0 = 0.01 m (b) Particle size d0 = 0.005 m (c) Particle size d0 = 0.0025 m

Figure 7: Pressure distributions for dam break analysis from different particle sizemodels d0 = 0.01, 0.005,and 0.0025m, respectively.


0 2 4 6 80

1000

2000

3000

4000

5000

6000

7000

8000

Pres

sure

(Pa)

Time (s)

d0 = 0.01 and α = 0.15d0 = 0.005 and α = 0.001d0 = 0.0025 and α = 0.00006

Figure 8: Time history for pressure distribution at different particle sizes with proper relaxation coefficient.

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

Exp. KoshizukaExp. M & M 1.125Exp. M & M 2.25

Cal. LeePresent

t (2g/L)0.5

Z/L

Figure 9: Comparison on dam break fronts in the dam break analysis.

0.6

m

1.2 m 2.02 m

Water A: 0.16 m

Figure 10: Dam break simulation corresponding to experiment by Zhou et al. [36].

the final stable state is compared between the proposed scheme and truly incompressiblescheme using the water height. It seems that the proposed scheme conserves the total volumecompared to the theoretical value of height about 0.22m, while the truly incompressible


t = 0.32 s, t(g/h)0.5 = 1.29

t = 0.76 s, t(g/h)0.5 = 3.07

t = 1.28 s, t(g/h)0.5 = 5.17

t = 1.68 s, t(g/h)0.5 = 6.78

t = 2.04 s, t(g/h)0.5 = 8.24

t = 8 s, t(g/h)0.5 = 32.33

t = 1.52 s, t(g/h)0.5 = 6.14

t = 2.4 s, t(g/h)0.5 = 9.69

(a) Stabilized ISPH model (α = 0.001) (b) Original ISPH model using α = 0

0 1100 2200 3300 4400 5500 0 1100 2200 3300(Pa)(Pa)

4400 5500

87776757473727177

−3−3 62 127 193 258 323

87776757473727177

−3−3 62 127 193 258 323

Figure 11: Pressure distribution for dam break analysis by (a) stabilized ISPH with eddy viscosity effect,and (b) original ISPH (α = 0) without eddy viscosity effect, respectively.


2 4 6 8 100

2000

4000

6000

Pres

sure

(Pa)

Experimental results Original ISPH

Stabilized ISPH Stabilized ISPH +

t(g/h)0.5

turbulence model

Figure 12: (a) Comparison between the current stabilized model including/excluding eddy viscosity,divergence-free scheme condition, and experimental data by Zhou et al. [36].

scheme cannot conserve the volume at the final stable state. Figure 12 shows the comparisonof pressure history at the right corner among our proposed scheme result with a properrelaxation coefficient, result from the truly incompressible scheme, and experimental data byZhou et al. [36]. Although the pressure level from the truly incompressible scheme (α = 0) islower than the experimental data in the entire simulation period, the evaluated pressure fromour proposed scheme shows a good agreement with the experimental data. In this figure,imaginary pressure peak is evaluated around t = 2.04 (t(g/h)0.5 = 8.24) in the results withoutturbulence model. The combination with the proposed stabilized ISPH and turbulence modelgenerates smoothed and accurate pressure distribution.

5.3. 3D Dam Break Flow with an Obstacle

The last application is one of the benchmark test suggested by SPH European ResearchInterest Community (SPHERIC). The experimental tests on a dam break flowwith an obstaclewas carried out at the Maritime Research Institute Netherlands (MARIN) as reported byKleefsman et al. [37].

Figures 13 and 14 show geometry of the experimental test and locations of pressuresensor, respectively. While ps1 to ps8 sensors were used in the experimental test, here onlyodd numbers of pressure sensor are utilized for the comparison. In the numerical modeling,the initial particle distance is fixed at 0.01m for both regions of water and wall. The totalnumber of particles is about 1.4 millions, and 0.67 million particles are located in the water. Inorder to evaluate an efficient relaxation coefficient, the same procedure as two-dimensionalcases is applied. First, the hydrostatic pressure test has been implemented by using the sameinitial particle distance d0 = 0.01m and time increment Δt = 0.001 s in the 3D dam breakproblem. Then the optimum relaxation coefficient α was fixed by 0.1.

The pressure time history on the front (ps1 and ps3) and top (ps5 and ps7) isshown in Figure 15. Figures 16 and 17 show the snapshots with particle pressure valuesand labels related to free surface, respectively. In Figure 16, the numerical solutions by our


Water

Water

Obstacle

0.161

0.29

50.

295

0.29

51

0.40

3

Ele

vati

on v

iew

Sect

ion

view

Obstacle0.161

1.228 1.248 0.744

Unit (m)0.

55

0.16

1

Figure 13: Geometry of the 3D dam break experiment.

0.176

0.176

0.403

0.16

1

0.161

0.0210.04

0.040.04

0.0210.021

0.04

0.04

0.04

0.021ps1

ps2

ps3

ps4

ps5

ps6

ps7

ps8

Figure 14: Locations of the pressure sensor on the obstacle.

proposed relaxed density invariant scheme (stabilizes ISPH) are compared with Kleefsman’sexperimental results and numerical results by the keeping divergence-free scheme withα = 0 (original ISPH). The first impact occurred at about 0.42 s both in the numerical andexperimental test, although the time of secondary hit has about 0.5 s difference (0.45 s and0.50 s, resp.). That is, our solution shows small delay as the time goes. The pressure resultingfrom the keeping divergence-free scheme shows lower value during the simulation, althougha smooth pressure distribution can be generated as in Figure 16. It seems that the keepingdivergence scheme cannot keep the total volume of water. On the other hand, except for thelocal pressure oscillation especially at ps5 and ps7, the pressure histories by our proposedscheme show good agreement with the experimental results.

Lee et al. [38] has been simulated to the same problem, and they have discussedthe difference between weakly compressible SPH and their proposed truly incompressibleSPH that is one of the keeping divergence-free scheme. According to their result, the weaklycompressible SPH shows a critical error in the pressure and their truly incompressible SPH


12000

10000

8000

6000

4000

2000

00 1 2 3 4 5 6 7

Time (s)

Pres

sure

(Pa)

(a) Output value at ps1

8000

6000

4000

2000

00 1 2 3 4 5 6 7

Time (s)

Pres

sure

(Pa)

(b) Output value at ps3

4000

3000

2000

1000

00 1 2 3 4 5 6 7

Time (s)

Pres

sure

(Pa)

Experimental dataOriginal ISPH

Stabilized ISPHStabilized ISPH+ Eddy viscosity

(c) Output value at ps5

4000

3000

2000

1000

00 1 2 3 4 5 6 7

Time (s)

Pres

sure

(Pa)

Experimental dataOriginal ISPH

Stabilized ISPHStabilized ISPH+ Eddy viscosity

(d) Output value at ps7

Figure 15: Time history of Pressure for 3D dam break at ps1, ps3, ps5, and ps7.

solution has the similar tendency as our results. However, the original ISPH scheme cannotkeep the total volume as far as we have checked.

6. Conclusion

A stabilized incompressible smoothed particle hydrodynamics is proposed to simulatefree surface flow. The modification is appeared in the source term of pressure Poissonequation, and the idea is similar to the recent development in Moving Particles Semi-implicit method (MPS). Although only one set of linear equations should be solved toevaluate pressure at each particle, both the velocity divergence-free condition and thedensity invariance condition can be approximately satisfied. The additional parameter is therelaxation coefficient, and its value can be calibrated by a simple hydrostatic simulation witha regular initial particle distribution. It has a uniform tendency that the relaxation coefficientbecomes smaller due to decrease in the initial particle distance. The efficiency and its accuracy


t = 0.42 s

= 0.7 st

t = 1 s

t = 1.5 s

t = 2 s

t = 4 s

t = 5 s

(a) (b)

Figure 16: Time sequence of 3D dam break simulation by (a) stabilized ISPH method with eddy viscosityeffect and (b) without eddy viscosity effect.


t = 0.42 s

t = 1 s

t = 4 s

t = 6 s(a) (b)

Figure 17: Time sequence for detection free surface in 3D dam break simulation by (a) stabilized ISPHmethod with eddy viscosity effect and (b) without eddy viscosity effect.

have been tested by the dam break in two- and three-dimension simulations compared totheir reference solutions. Our proposed scheme shows the clear advantage to keep the totalvolume by comparing the original ISPH, and it may contribute to have an accurate pressurevalue. However, it still has an artificial oscillation in the pressure value with original viscosity.The additional viscosity based on the Subparticle Scale turbulence model shows an importantrole to generate smoother pressure distribution and to decrease the number of isolatedparticles after the splash in the dam break problems.

Acknowledgments

The first author is partially supported by the Ministry of Education, Science, Sports andCulture of Japan (Grant-in-Aid for Young Scientist (B) 2176036).

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